It would be nice if everyone were to acquire a solid grounding in mathematics as part of their education. It would also be nice if the body of mathematical knowledge remained the same year after year. But neither of these assumptions is true or realistic.

Some tools have come in to being that make the practice of mathematics easier and more rewarding. Beginning in the middle of the last century, electronic computers began to change how numerical mathematics is practiced. More recently, programs capable of symbolic operations have changed mathematics in the most fundamental way.

In 1976, Appel and Haken published a computer-based proof of the four-color map theorem, the first proof that required a computer (other proofs had been aided by, but did not require, computer assistance). This theorem states that only four colors are required to color a map so that no two adjacent territories share the same color. The problem was originally posed in 1852, but its proof proved too complex for the skills of mathematicians without computers.

This proof was not accepted by all mathematicians on the ground that no one has managed to check the proof entirely without computer assistance. In fact, because of the complexity of the problem, most efforts to verify the computer proof have also used computers.

At the time of writing, computers have become more widely accepted in mathematical practice, but with occasional appropriate criticism, mostly questioning the rigor of the results (and of the practitioners). Others argue that people will be more likely to learn advanced mathematical topics because computers will handle the more menial tasks and present the results in a way likely to help reveal the underlying concepts.

The most important recent development in computer mathematics are programs like Mathematica, programs able to perform symbolic as well as numeric mathematics. These programs greatly speed both the learning and practice of mathematics, again with the caution that this may be at the sacrifice of deep understanding and an overly trusting stance toward the computer's results. Most of the results and graphics in this page set were created using Mathematica.

Here is an example of an input to Mathematica. It is the entry of a differential equation from this page set — it is the second form of a very common physical equation discussed here:

DSolve[y[t] + r c y'[t] == m Sin[&omega t], y, t]

That's all that is required. In one additional step, the result is encapsulated in a suitable function, ready for application. If a differential equation is not soluble in closed form, the program will let you know, and there is the option of using a numerical differential equation solver with the name "NDSolve".

The argument that symbolic mathematical programs can lead to intellectual laziness may have merit, but I personally know a great deal more mathematics than I did before the advent of this class of program, and I understand and solve many more kinds of problems that I would have considered tackling before they appeared.

Because there are now a reasonable number of competing, powerful symbolic mathematics programs, it is hoped that this will result in price pressure and more reasonable prices in the future.

Update, 2007: for students, and for those who would like to experiment with symbolic algebra without spending thousands of dollars, I recommend the free program Maxima. Read this article for more on Maxima.

Here is a list of the more common symbolic mathematics programs suitable for end-user computers:

Name	Link	Platforms	Cost full/student (2007), US$	Comment
Maxima	http://maxima.sourceforge.net/	Windows, Linux	Free	Steeper learning curve compared to more expensive packages, but highly recommended based on price. For more, read my expository article and tutorial.
Mathematica	http://www.wolfram.com/	Windows, Macintosh, Linux	$2495/$140	Powerful, very expensive. A reasonably priced student edition is available. Used extensively by author.
Maple	http://www.maplesoft.com/	Windows, Macintosh, Linux	$1995/$99	Appears to be powerful, very expensive. Student price available.
MuPad	http://www.mupad.com/	Windows, Linux	Was once free for non-commercial users, but no more.	Seems capable. Linux version not up to date.
MathCAD	http://www.mathcad.com/	Windows	$1195/$1195	Windows only.

Acknowledgments

Much of the preliminary mathematical work, and most of the graphs and graphic equation renderings, were performed in Mathematica. The computer graphic images were rendered by POV-Ray with the help of KPovModeler.